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Properties

Label 2-1110-185.159-c1-0-28
Degree $2$
Conductor $1110$
Sign $0.539 + 0.841i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + (−0.696 − 2.12i)5-s − 0.999i·6-s + (1.07 + 0.619i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (1.49 − 1.66i)10-s + 2.37·11-s + (0.866 − 0.499i)12-s + (1.12 − 1.94i)13-s + 1.23i·14-s + (−0.459 + 2.18i)15-s + (−0.5 − 0.866i)16-s + (0.499 + 0.865i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.311 − 0.950i)5-s − 0.408i·6-s + (0.405 + 0.233i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.471 − 0.526i)10-s + 0.717·11-s + (0.249 − 0.144i)12-s + (0.311 − 0.538i)13-s + 0.330i·14-s + (−0.118 + 0.565i)15-s + (−0.125 − 0.216i)16-s + (0.121 + 0.209i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 + 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.539 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.539 + 0.841i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.539 + 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.311160627\)
\(L(\frac12)\) \(\approx\) \(1.311160627\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (0.696 + 2.12i)T \)
37 \( 1 + (-5.14 - 3.25i)T \)
good7 \( 1 + (-1.07 - 0.619i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 2.37T + 11T^{2} \)
13 \( 1 + (-1.12 + 1.94i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.499 - 0.865i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (7.10 + 4.10i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.09T + 23T^{2} \)
29 \( 1 + 5.72iT - 29T^{2} \)
31 \( 1 + 3.10iT - 31T^{2} \)
41 \( 1 + (-4.36 + 7.55i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 6.26T + 43T^{2} \)
47 \( 1 + 6.29iT - 47T^{2} \)
53 \( 1 + (-10.0 + 5.82i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.06 + 0.617i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.52 + 2.61i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.63 - 0.944i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.46 + 4.26i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.123iT - 73T^{2} \)
79 \( 1 + (13.2 + 7.66i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.36 - 3.67i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.869 + 0.502i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.95T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.487453857622442970744447964023, −8.608834096982185235693614598078, −8.129731174231044041006910331315, −7.15115445244303992391242315479, −6.19400308261448392635266232008, −5.54454941051254953156818538178, −4.54176463958548045159796897973, −3.94007487610436145242303885117, −2.16592852322758073864358614897, −0.58386586634895136535873559092, 1.44327579086393163732625489406, 2.77607800858589312107388985164, 4.02357970503951208776824777641, 4.35319891110665422222132005299, 5.80865418189900524328652318447, 6.42987504511004462225232019511, 7.34077043598732338375990234083, 8.434741224029335065968102270970, 9.384468494472074978741043806638, 10.26761544833651039195213403778

Graph of the $Z$-function along the critical line